METHOD OF LAGRANGE 108 Derived Functions 108 An extension of ordinary Analysis 108 _Example_: Tangents 109 _Fundamental Identity of the three Methods_ 110 _Their comparative Value_ 113 That of Leibnitz 113 That of Newton 115 That of Lagrange 117 CHAPTER IV. Page THE DIFFERENTIAL AND INTEGRAL CALCULUS 120 ITS TWO FUNDAMENTAL DIVISIONS 120 THEIR RELATIONS TO EACH OTHER 121 1. Use of the Differential Calculus as preparatory to that of the Integral 123 2. Employment of the Differential Calculus alone 125 3. Employment of the Integral Calculus alone 125 Three Classes of Questions hence resulting 126 THE DIFFERENTIAL CALCULUS 127 Two Cases: Explicit and Implicit Functions 127 Two sub-Cases: a single Variable or several 129 Two other Cases: Functions separate or combined 130 Reduction of all to the Differentiation of the ten elementary Functions 131 Transformation of derived Functions for new Variables 132 Different Orders of Differentiation 133 Analytical Applications 133 THE INTEGRAL CALCULUS 135 Its fundamental Division: Explicit and Implicit Functions 135 Subdivisions: a single Variable or several 136 Calculus of partial Differences 137 Another Subdivision: different Orders of Differentiation 138 Another equivalent Distinction 140 _Quadratures_ 142 Integration of Transcendental Functions 143 Integration by Parts 143 Integration of Algebraic Functions 143 Singular Solutions 144 Definite Integrals 146 Prospects of the Integral Calculus 148 CHAPTER V. Page THE CALCULUS OF VARIATIONS 151 PROBLEMS GIVING RISE TO IT 151 Ordinary Questions of Maxima and Minima 151 A new Class of Questions 152 Solid of least Resistance; Brachystochrone; Isoperimeters 153 ANALYTICAL NATURE OF THESE QUESTIONS 154 METHODS OF THE OLDER GEOMETERS 155